Question: The graph of the line $x+y=b$ is a perpendicular bisector of the line segment from $(1,3)$ to $(5,7)$. What is the value of b?
Explanation: If the line $x+y=b$ is the perpendicular bisector of the segment from $(1,3)$ to $(5,7)$, it must pass through the midpoint of this segment.  The midpoint is: $$\left(\frac{1+5}{2},\frac{3+7}{2}\right)=(3,5)$$This point lies on the line $x+y=b$, so we must have $3+5=b\Rightarrow b=\boxed{8}$.